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Dissipative dynamics of topological defects in the spiral phase
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 310 (2007) e523–e524 www.elsevier.com/locate/jmmm
Dissipative dynamics of topological defects in the spiral phase P. Rusek Institute of Physics, Wroclaw Institute of Technology, 50-370 Wroclaw, Poland Available online 7 November 2006
Abstract The equations of motion for topological defects in the spiral phase of cuprates are derived using a concept of gauge fields. We have found that the dynamics of topological defects is pure dissipative. Assuming that the charge carriers are attached to the topological defects the in-plane resistivity was evaluated and we have found it is proportional to temperature in agreement with experiment. r 2006 Elsevier B.V. All rights reserved. PACS: 75.10.Nr; 74.25.Fy; 74.72.Dn Keywords: Spiral phase; Topological defects; Magnons; Gauge field
In Nd doped La2x Srx CuO4 ð0:02oxo0:05Þ (LCO) the holes induce magnetic dipole moment which causes the formation of the spiral phase in spins system. Moreover, the long range dipole potential favorites frustrated spin configurations. The spiral magnetic order allows the existence of topological defects which are related to the first homotopy group of SOð3Þ group, p1 ðSOð3ÞÞ ¼ Z 2 (hereafter we refer to them as TD). The ground state spin configuration of LCO is therefore a set of random spirals with well-developed TD with the charge carriers attached to them [1]. In this scenario, the charge transport in LCO can be considered in terms of the TD motion [2]. Using basic commutation relation for spin and the TD momentum we shall give strong argument that the TD motion is pure dissipative. The evaluated, from the Fermi golden rule, a damping coefficient of TD, at high temperature, leads to linear temperature dependence of the inverse transport relaxation time t1 of TD, thus to linear temperature dependence of the LCO resistivity as observed in experiment [3]. Our model indicates that the dissipative motion of TD follows from the specific kinematics of magnetic moment but the temperature dependence LCO resistivity does not depend, at high temperature, on the specific form of the cross-section of magnons scattered on TD. Compare to others models of TD dynamics in LCO [2] Tel.: +48 71 3303057; fax.: +48 71 3211235.
E-mail address: [email protected]. 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.449
our phenomenological (continuum) model works for any shape of TD (bounded and unbounded), it exposes and explores the general features of TD dynamics regardless of background, and it evaluates the temperature dependence of a TD transport relaxation time in the framework of kinetic theory that is commonly used procedure in magnetism [4]. The rotational symmetry of the spiral ground state of LCO is completely broken, i.e. its energy depends on any rotation of spin space. In that case the order parameter has SOð3Þ symmetry, hence it allows the existence of the TDZ2 vortices which are the points (in 2D) (the lines in 3D) where the order parameter is singular. Being interested in semiclassical limit as the order parameter we take the three mutually orthonormal vectors lk ; k ¼ 1; 2; 3, that is locally equivalent to a rotation matrix O. In the presence of TD the change of order parameter due to the change of x,dO requires to introduce an additional quantity the connection akm , which in the absence of TD akm ¼ 1=2ek;a;b ðOqm O1 Þa;b ’ qm jk (the upper(lower) indices indicate components in spin(real) space). Here j is the parameter of rotation group Oð3Þ (ðj1 ; j2 ; j3 Þ j ¼ n tan y=2, and y is the rotation angle about an axes parallel to unit vector n). In the presence of TD dO ¼ dO þ ½a^ m ; O dxm , where a^ m ¼ alm t^l ; t^l are the generators of SOð3Þ group. dO is integrated part of dO. In fact akm is Yang-Mills gauge potential of SOð3Þ group. Since the topological charge of TD is conserved we can interpret j km ¼ F km;0 as the density of TD current and
ARTICLE IN PRESS e524
P. Rusek / Journal of Magnetism and Magnetic Materials 310 (2007) e523–e524
rka ¼ ea;m;n F km;n (m; n ¼ 1; 2Þ as the density of TD ðF km;n ¼ qm akn qn akm þ ðam an Þk ; k ¼ 1; 2; 3; m; n ¼ 0; 1; 2Þ [5]. To describe the motion of TD we apply the (quantum) brownian motion concept [6]. A TD we consider as a classical particle, with large mass M, which interact with a bath of magnons when it is moving. The equations of motion for TD and magnons we shall derive in hydrodynamic approximation using Hamilton formalism. The equations of motion in that case are Liouville equations _ n ¼ fH; An g dR ¼ dH fAi ; An g dR , A dH dH dðdA dðdA Þ dAi Þ n n
(1)
where it is assumed H depends on the quantities Ai ; i ¼ 1; . . . ; N and R is a dissipation function. The total momentum of a system pt is a sum of TD p and magnons’ pm momentum. The last one can be taken as the momentum of the vector field j i.e. pmm ¼ ðqL=qj_ l Þrm jl ’ S l rm jl ’ S l alm where S is the spin of j field. Quantizing j pm may be interpreted as a momentum of magnons which interact with a moved TD. The dynamical variables which determine a TD dynamics are a mass density of TD r (required in hydrodynamical description), pm , coupled to it am ’ rm j and S which describes magnons. In that case H ¼ Hðpm ; r; am ; SÞ hence the Poisson brackets between pm ; r; am ; S are needed. The non-vanishing Poisson brackets, in our case, are [5] fMk ðxÞ; Ml ðx0 Þg ¼ eklm Mm ðxÞdðx x0 Þ, fpb ðxÞ; ala ðx0 Þg ¼ F lba dðx x0 Þ, fpa ðxÞ; pb ðx0 Þg ¼ ðpb ðxÞrxa pa ðx0 Þrx0 b þ Ml F lab Þdðx x0 Þ, ð2Þ
The Hamiltonian of our system can be taken in the form H¼
1 1 1 ðp pm Þ2 þ M2 þ Rs a2m , 2R t 2w 2
½ðnq þ 1Þðnp þ 1Þnp0 nq0 np nq ðnq0 þ 1Þðnp0 þ 1Þ Dðp þ q p0 q0 Þdðep þ ep ep0 eq0 Þ.
0 fMl ðxÞ; aka ðx0 Þg ¼ 12dlk rx dðx x0 Þ elkm am a ðxÞdðx x Þ,
fpa ðxÞ; Rðx0 Þg ¼ Rrxa dðx x0 Þ.
equation of motion for p and R are [5] dpa ¼ Ml F lga vg þ rs F lab alb qb ðpa vb Þ dt dH pb qa v b r 2 v a qa r , dr dr ð5Þ ¼ qb ðvb rÞ. dt In linear approximation Eqs. (5) yield p_a r2 va ¼ r1 pa , i.e. the motion of TD is pure dissipative. The reason of the dissipative TD motion is not only the friction of TD by other degrees of freedom but also its interaction with magnons. The last effect is represented by the first term (nonlinear) in the first equation of Eqs. (5). In the second quantization representation(using magnons creation and annihilation operators) that term describes the scattering of magnons on TD which moves with the velocity v. The damping of TD Gp due to the magnon scattering on it we evaluate from the amplitude Fðp; p0 ; q; q0 Þ of the process: the magnon and TD with initial momentums q; p are scattered to the states with momentums q0 ; p0 , respectively, [4,7] Gp ¼ ðdIðnp Þ=dnp Þ, where Iðnp Þ is a collision integral. Iðnp Þ can be evaluated using Fermi’s golden rule [4,7] 16p X Iðnp Þ ¼ jFðp; p0 ; q; q0 Þj2 _ p0 ;q;q0
(3)
which is the sum of the kinetic energy of TD and magnons. The last term of H in Eq. (3) is not gauge invariant. Its value differs from a gauge invariant term, where a covariant derivative of j should be taken instead aa , by terms which are higher order in fields (in our gauge) and therefore they can be neglected in our case. In other words, the gauge symmetry breaking of H by the last term in H does not change final results qualitatively (and quantitatively in linear approximation). In the same approximation the dissipation function can be taken in the form 1 dH 2 dH 2 2 R ¼ r1 ðrl SÞ þ pr þ r2 . (4) 2 dam dp The first term in R describes spin diffusion, the second the dissipation in TD, the last one takes into account a friction of TD caused by its interaction with other degrees of freedom. The equations of motion Eq. (1) for Ma and aa are decoupled, in linear approximation, and they yield the magnons with linear dispersion o2a ðqÞ ¼ rs w1 q2 [5,7]. The
Here nq ; nq0 and np ; np0 are Bose and Boltzmann occupation numbers, respectively. The evaluation of Gp , as the function of p, T is not trivial task [4,7]. However at high temperature the temperature dependence of Gp can be found relatively easily, independently on the scattering amplitude Fðp; p0 ; q; q0 Þ. Using analogous calculations as in Refs. [4,8] we have found that at high temperature, o0 5T, (o0 is the cutoff of magnons energy) to the lowest order in o0 =T51, Gp T. It means that the transport relaxation time t as well as the mobility m of disclinations are proportional to T 1 at high temperature. Assuming that the charge carriers are attached to TD we conclude that the resistivity of LCO rr m1 t1 T as observed in experiment [3]. Our model works also for an anisotropic spiral phase with an external magnetic field allowance. The TD dynamics is still dissipative in that case but the resistivity is proportional to the temperature (at high temperature). References [1] N. Hasselmann, et al., Phys. Rev. B 69 (2004) 014424. [2] V. Juricic, et al., Phys. Rev. Lett. 92 (2004) 137202. [3] Y. Ando, et al., Phys. Rev. Lett. 87 (2001) 017001; Y. Ando, et al., Phys. Rev. Lett. 88 (2002) 137005. [4] A.I. Akhiezier, V.G. Bariakhtar, S.V. Peletminskii, Spin Waves, Pergamon, New York, 1972. [5] I.E. Dzyaloshinskii, G.E. Volovik, Ann. Phys. 125 (1980) 67. [6] A.H. Castro Neto, A.O. Caldeira, Phys. Rev. Lett. 67 (1991) 1960. [7] P. Rusek, J. Phys. C 14 (1981) 3769. [8] H.B. Schuttler, T. Holstein, Phys. Rev. Lett. 51 (1983) 2337.
Journal of Magnetism and Magnetic Materials 310 (2007) e523–e524 www.elsevier.com/locate/jmmm
Dissipative dynamics of topological defects in the spiral phase P. Rusek Institute of Physics, Wroclaw Institute of Technology, 50-370 Wroclaw, Poland Available online 7 November 2006
Abstract The equations of motion for topological defects in the spiral phase of cuprates are derived using a concept of gauge fields. We have found that the dynamics of topological defects is pure dissipative. Assuming that the charge carriers are attached to the topological defects the in-plane resistivity was evaluated and we have found it is proportional to temperature in agreement with experiment. r 2006 Elsevier B.V. All rights reserved. PACS: 75.10.Nr; 74.25.Fy; 74.72.Dn Keywords: Spiral phase; Topological defects; Magnons; Gauge field
In Nd doped La2x Srx CuO4 ð0:02oxo0:05Þ (LCO) the holes induce magnetic dipole moment which causes the formation of the spiral phase in spins system. Moreover, the long range dipole potential favorites frustrated spin configurations. The spiral magnetic order allows the existence of topological defects which are related to the first homotopy group of SOð3Þ group, p1 ðSOð3ÞÞ ¼ Z 2 (hereafter we refer to them as TD). The ground state spin configuration of LCO is therefore a set of random spirals with well-developed TD with the charge carriers attached to them [1]. In this scenario, the charge transport in LCO can be considered in terms of the TD motion [2]. Using basic commutation relation for spin and the TD momentum we shall give strong argument that the TD motion is pure dissipative. The evaluated, from the Fermi golden rule, a damping coefficient of TD, at high temperature, leads to linear temperature dependence of the inverse transport relaxation time t1 of TD, thus to linear temperature dependence of the LCO resistivity as observed in experiment [3]. Our model indicates that the dissipative motion of TD follows from the specific kinematics of magnetic moment but the temperature dependence LCO resistivity does not depend, at high temperature, on the specific form of the cross-section of magnons scattered on TD. Compare to others models of TD dynamics in LCO [2] Tel.: +48 71 3303057; fax.: +48 71 3211235.
E-mail address: [email protected]. 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.449
our phenomenological (continuum) model works for any shape of TD (bounded and unbounded), it exposes and explores the general features of TD dynamics regardless of background, and it evaluates the temperature dependence of a TD transport relaxation time in the framework of kinetic theory that is commonly used procedure in magnetism [4]. The rotational symmetry of the spiral ground state of LCO is completely broken, i.e. its energy depends on any rotation of spin space. In that case the order parameter has SOð3Þ symmetry, hence it allows the existence of the TDZ2 vortices which are the points (in 2D) (the lines in 3D) where the order parameter is singular. Being interested in semiclassical limit as the order parameter we take the three mutually orthonormal vectors lk ; k ¼ 1; 2; 3, that is locally equivalent to a rotation matrix O. In the presence of TD the change of order parameter due to the change of x,dO requires to introduce an additional quantity the connection akm , which in the absence of TD akm ¼ 1=2ek;a;b ðOqm O1 Þa;b ’ qm jk (the upper(lower) indices indicate components in spin(real) space). Here j is the parameter of rotation group Oð3Þ (ðj1 ; j2 ; j3 Þ j ¼ n tan y=2, and y is the rotation angle about an axes parallel to unit vector n). In the presence of TD dO ¼ dO þ ½a^ m ; O dxm , where a^ m ¼ alm t^l ; t^l are the generators of SOð3Þ group. dO is integrated part of dO. In fact akm is Yang-Mills gauge potential of SOð3Þ group. Since the topological charge of TD is conserved we can interpret j km ¼ F km;0 as the density of TD current and
ARTICLE IN PRESS e524
P. Rusek / Journal of Magnetism and Magnetic Materials 310 (2007) e523–e524
rka ¼ ea;m;n F km;n (m; n ¼ 1; 2Þ as the density of TD ðF km;n ¼ qm akn qn akm þ ðam an Þk ; k ¼ 1; 2; 3; m; n ¼ 0; 1; 2Þ [5]. To describe the motion of TD we apply the (quantum) brownian motion concept [6]. A TD we consider as a classical particle, with large mass M, which interact with a bath of magnons when it is moving. The equations of motion for TD and magnons we shall derive in hydrodynamic approximation using Hamilton formalism. The equations of motion in that case are Liouville equations _ n ¼ fH; An g dR ¼ dH fAi ; An g dR , A dH dH dðdA dðdA Þ dAi Þ n n
(1)
where it is assumed H depends on the quantities Ai ; i ¼ 1; . . . ; N and R is a dissipation function. The total momentum of a system pt is a sum of TD p and magnons’ pm momentum. The last one can be taken as the momentum of the vector field j i.e. pmm ¼ ðqL=qj_ l Þrm jl ’ S l rm jl ’ S l alm where S is the spin of j field. Quantizing j pm may be interpreted as a momentum of magnons which interact with a moved TD. The dynamical variables which determine a TD dynamics are a mass density of TD r (required in hydrodynamical description), pm , coupled to it am ’ rm j and S which describes magnons. In that case H ¼ Hðpm ; r; am ; SÞ hence the Poisson brackets between pm ; r; am ; S are needed. The non-vanishing Poisson brackets, in our case, are [5] fMk ðxÞ; Ml ðx0 Þg ¼ eklm Mm ðxÞdðx x0 Þ, fpb ðxÞ; ala ðx0 Þg ¼ F lba dðx x0 Þ, fpa ðxÞ; pb ðx0 Þg ¼ ðpb ðxÞrxa pa ðx0 Þrx0 b þ Ml F lab Þdðx x0 Þ, ð2Þ
The Hamiltonian of our system can be taken in the form H¼
1 1 1 ðp pm Þ2 þ M2 þ Rs a2m , 2R t 2w 2
½ðnq þ 1Þðnp þ 1Þnp0 nq0 np nq ðnq0 þ 1Þðnp0 þ 1Þ Dðp þ q p0 q0 Þdðep þ ep ep0 eq0 Þ.
0 fMl ðxÞ; aka ðx0 Þg ¼ 12dlk rx dðx x0 Þ elkm am a ðxÞdðx x Þ,
fpa ðxÞ; Rðx0 Þg ¼ Rrxa dðx x0 Þ.
equation of motion for p and R are [5] dpa ¼ Ml F lga vg þ rs F lab alb qb ðpa vb Þ dt dH pb qa v b r 2 v a qa r , dr dr ð5Þ ¼ qb ðvb rÞ. dt In linear approximation Eqs. (5) yield p_a r2 va ¼ r1 pa , i.e. the motion of TD is pure dissipative. The reason of the dissipative TD motion is not only the friction of TD by other degrees of freedom but also its interaction with magnons. The last effect is represented by the first term (nonlinear) in the first equation of Eqs. (5). In the second quantization representation(using magnons creation and annihilation operators) that term describes the scattering of magnons on TD which moves with the velocity v. The damping of TD Gp due to the magnon scattering on it we evaluate from the amplitude Fðp; p0 ; q; q0 Þ of the process: the magnon and TD with initial momentums q; p are scattered to the states with momentums q0 ; p0 , respectively, [4,7] Gp ¼ ðdIðnp Þ=dnp Þ, where Iðnp Þ is a collision integral. Iðnp Þ can be evaluated using Fermi’s golden rule [4,7] 16p X Iðnp Þ ¼ jFðp; p0 ; q; q0 Þj2 _ p0 ;q;q0
(3)
which is the sum of the kinetic energy of TD and magnons. The last term of H in Eq. (3) is not gauge invariant. Its value differs from a gauge invariant term, where a covariant derivative of j should be taken instead aa , by terms which are higher order in fields (in our gauge) and therefore they can be neglected in our case. In other words, the gauge symmetry breaking of H by the last term in H does not change final results qualitatively (and quantitatively in linear approximation). In the same approximation the dissipation function can be taken in the form 1 dH 2 dH 2 2 R ¼ r1 ðrl SÞ þ pr þ r2 . (4) 2 dam dp The first term in R describes spin diffusion, the second the dissipation in TD, the last one takes into account a friction of TD caused by its interaction with other degrees of freedom. The equations of motion Eq. (1) for Ma and aa are decoupled, in linear approximation, and they yield the magnons with linear dispersion o2a ðqÞ ¼ rs w1 q2 [5,7]. The
Here nq ; nq0 and np ; np0 are Bose and Boltzmann occupation numbers, respectively. The evaluation of Gp , as the function of p, T is not trivial task [4,7]. However at high temperature the temperature dependence of Gp can be found relatively easily, independently on the scattering amplitude Fðp; p0 ; q; q0 Þ. Using analogous calculations as in Refs. [4,8] we have found that at high temperature, o0 5T, (o0 is the cutoff of magnons energy) to the lowest order in o0 =T51, Gp T. It means that the transport relaxation time t as well as the mobility m of disclinations are proportional to T 1 at high temperature. Assuming that the charge carriers are attached to TD we conclude that the resistivity of LCO rr m1 t1 T as observed in experiment [3]. Our model works also for an anisotropic spiral phase with an external magnetic field allowance. The TD dynamics is still dissipative in that case but the resistivity is proportional to the temperature (at high temperature). References [1] N. Hasselmann, et al., Phys. Rev. B 69 (2004) 014424. [2] V. Juricic, et al., Phys. Rev. Lett. 92 (2004) 137202. [3] Y. Ando, et al., Phys. Rev. Lett. 87 (2001) 017001; Y. Ando, et al., Phys. Rev. Lett. 88 (2002) 137005. [4] A.I. Akhiezier, V.G. Bariakhtar, S.V. Peletminskii, Spin Waves, Pergamon, New York, 1972. [5] I.E. Dzyaloshinskii, G.E. Volovik, Ann. Phys. 125 (1980) 67. [6] A.H. Castro Neto, A.O. Caldeira, Phys. Rev. Lett. 67 (1991) 1960. [7] P. Rusek, J. Phys. C 14 (1981) 3769. [8] H.B. Schuttler, T. Holstein, Phys. Rev. Lett. 51 (1983) 2337.